AMC 10 · 2010 · #6

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
inscribed-angleisosceles-triangleangle-sum-triangle convert-to-algebra ↑ Prerequisites: angle-sum-triangle
📏 Short solution 💡 2 insights
Problem
A circle has center O, and AB is a diameter, so A, O, and B lie on one straight line. A third point C sits on the circle, and the central angle ∠ COB measures 50°. We want the measure of ∠ CAB, the angle at vertex A inside triangle ABC.

Pick an answer.

(A)
20
(B)
25
(C)
45
(D)
50
(E)
65

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The angle we want, ∠ CAB, is one of the two base angles of triangle OAC. The clean move is Tool #4 (Introduce a Variable): call that base angle x. Because OA and OC are both radii, triangle OAC is isosceles, so the OTHER base angle is also x for free — one variable captures two angles. Tool #1 (Draw a Diagram) makes the picture do the heavy lifting: drawing radius OC reveals the isosceles triangle, and seeing A, O, B on one line reveals the straight-angle relationship. Tool #7 (Identify Subproblems) splits the job into two easy pieces — first pin down ∠ AOC from the straight line, then use the triangle's angle sum to solve for x.

1STEP 1

Name the base angle x

Draw radius OC. Triangle OAC has OA = OC, so it is isosceles: let x = ∠ CAB, and then ∠ OCA = x too.

OA = OC → ∠ OCA = ∠ OAC = x
2STEP 2

Use the straight line at O

AB is a diameter, so ∠ AOC and ∠ COB fill the straight angle at O: ∠ AOC = 180° - 50° = 130°.

∠ AOC = 180° - 50° = 130°
3STEP 3

Solve the angle-sum equation

The angles of triangle OAC sum to 180°, so x + x + 130° = 180°, giving 2x = 50° and ∠ CAB = 25°.

x + x + 130° = 180° → 2x = 50° → x = 25°
Answer
25
Check against the inscribed-angle idea. ∠ CAB is an inscribed angle standing on arc BC, and ∠ COB is the central angle standing on the same arc. An inscribed angle is always half of the central angle on the same arc, so ∠ CAB = 1/2(50°) = 25° — exactly what we found. It is also sensible that ∠ CAB = 25° is smaller than the central angle 50°. This matches (B).
💡Key takeaway

Two radii make an isosceles triangle, so its two bottom angles are equal; find the top angle from the straight line, and the leftover splits in half to give 25°.

  • Name the base angle x
  • Use the straight line at O
  • Solve the angle-sum equation

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